Numerical Methods For Stochastic Ordinary Differential .

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsNumerical Methods for Stochastic OrdinaryDifferential Equations (SODEs)Josh BuliGraduate Student SeminarUniversity of California, RiversideApril 1, 2016HO Methods

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsIntroductionDeterministic ODEs vs. Stochastic Differential EquationsHO Methods

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsIntroductionDeterministic ODEs vs. Stochastic Differential EquationsBrownian Motion and Wiener Process12Definitions, Properties, ExamplesSample Paths in R, R2 , R3HO Methods

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsIntroductionDeterministic ODEs vs. Stochastic Differential EquationsBrownian Motion and Wiener Process12Definitions, Properties, ExamplesSample Paths in R, R2 , R3Stochastic Calculus1Itô and Stratonovich CalculusHO Methods

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsIntroductionDeterministic ODEs vs. Stochastic Differential EquationsBrownian Motion and Wiener Process12Definitions, Properties, ExamplesSample Paths in R, R2 , R3Stochastic Calculus1Itô and Stratonovich CalculusGeometric Brownian MotionHO Methods

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsIntroductionDeterministic ODEs vs. Stochastic Differential EquationsBrownian Motion and Wiener Process12Definitions, Properties, ExamplesSample Paths in R, R2 , R3Stochastic Calculus1Itô and Stratonovich CalculusGeometric Brownian MotionEuler-Maruyama MethodMilstein MethodMonte Carlo Method123What is a Monte Carlo Simulation?Approximation of Logistic EquationApproximation of Geometric Brownian MotionHO Methods

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsIntroductionDeterministic ODEs vs. Stochastic Differential EquationsBrownian Motion and Wiener Process12Definitions, Properties, ExamplesSample Paths in R, R2 , R3Stochastic Calculus1Itô and Stratonovich CalculusGeometric Brownian MotionEuler-Maruyama MethodMilstein MethodMonte Carlo Method123What is a Monte Carlo Simulation?Approximation of Logistic EquationApproximation of Geometric Brownian MotionHigher Order Taylor and Runge Kutta MethodsHO Methods

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsHO MethodsDifferential EquationsMotivationWe would like to study processes or systems that are driven bynoise, or have uncertainty in coefficients.

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsHO MethodsDifferential EquationsMotivationWe would like to study processes or systems that are driven bynoise, or have uncertainty in coefficients.SDEs arise in modeling stock prices, thermal fluctuations,mathematical biology, etc.1Geometric Brownian motion, Langevin equation,Ornstein-Uhlenbeck process (Fokker-Planck equation), etc.

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsHO MethodsDifferential EquationsMotivationWe would like to study processes or systems that are driven bynoise, or have uncertainty in coefficients.SDEs arise in modeling stock prices, thermal fluctuations,mathematical biology, etc.1Geometric Brownian motion, Langevin equation,Ornstein-Uhlenbeck process (Fokker-Planck equation), etc.Stochastic terms also arise in PDEs as well.1Laplace, heat, wave equations with white noise forcing,stochastic Burgers’ equation, KPZ equation, etc.

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsHO MethodsDifferential EquationsMotivationWe would like to study processes or systems that are driven bynoise, or have uncertainty in coefficients.SDEs arise in modeling stock prices, thermal fluctuations,mathematical biology, etc.1Geometric Brownian motion, Langevin equation,Ornstein-Uhlenbeck process (Fokker-Planck equation), etc.Stochastic terms also arise in PDEs as well.1Laplace, heat, wave equations with white noise forcing,stochastic Burgers’ equation, KPZ equation, etc.Applications include population dynamics, neuron activity,option pricing, radio-astronomy, satellite orbit stability, bloodclotting, turbulent diffusion, Josephson tunneling insemiconductors, stochastic differential geometry, and manymore.

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsHO MethodsDifferential EquationsMotivationWe would like to study processes or systems that are driven bynoise, or have uncertainty in coefficients.SDEs arise in modeling stock prices, thermal fluctuations,mathematical biology, etc.1Geometric Brownian motion, Langevin equation,Ornstein-Uhlenbeck process (Fokker-Planck equation), etc.Stochastic terms also arise in PDEs as well.1Laplace, heat, wave equations with white noise forcing,stochastic Burgers’ equation, KPZ equation, etc.Applications include population dynamics, neuron activity,option pricing, radio-astronomy, satellite orbit stability, bloodclotting, turbulent diffusion, Josephson tunneling insemiconductors, stochastic differential geometry, and manymore.Filtering problems - algorithms that use measurements overtime that contain “noise”, and give estimates for unknown

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsDifferential EquationsDeterministic ODEsConsider the ordinary differential equation(ẋ(t) f(t, x(t))x(0) x0for t 0x0 Rnwhere f is a given smooth vector field, and the solutionx(t) : [0, ) Rn is the trajectory.HO Methods

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsHO MethodsDifferential EquationsDeterministic ODEsConsider the ordinary differential equation(ẋ(t) f(t, x(t))x(0) x0for t 0x0 Rnwhere f is a given smooth vector field, and the solutionx(t) : [0, ) Rn is the trajectory. Under some regularityassumptions on the vector field f, the above ODE has a solutionthat is uniquely determined by the initial condition x0 . Oneexample we will see later is the logistic equation,

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsHO MethodsDifferential EquationsDeterministic ODEsConsider the ordinary differential equation(ẋ(t) f(t, x(t))x(0) x0for t 0x0 Rnwhere f is a given smooth vector field, and the solutionx(t) : [0, ) Rn is the trajectory. Under some regularityassumptions on the vector field f, the above ODE has a solutionthat is uniquely determined by the initial condition x0 . Oneexample we will see later is the logistic equation,(ẋ(t) x(t)(1 x(t))x(0) x0which has the exact solution x(t) for t 0x0 R 1 1 1 1x0e t

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsHO MethodsDifferential EquationsStochastic ODEsHow do we introduce randomness into the general ODE on theprevious slide? First, we need some definitions to make sense ofwhat a solution to a SDE is.

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsHO MethodsDifferential EquationsStochastic ODEsHow do we introduce randomness into the general ODE on theprevious slide? First, we need some definitions to make sense ofwhat a solution to a SDE is.DefinitionLet Ω be a non-empty set, U be a σ-algebra of subsets of Ω, and Pbe the probability measure on U. We define a probability spaceto be the triple (Ω, U, P).

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsHO MethodsDifferential EquationsStochastic ODEsHow do we introduce randomness into the general ODE on theprevious slide? First, we need some definitions to make sense ofwhat a solution to a SDE is.DefinitionLet Ω be a non-empty set, U be a σ-algebra of subsets of Ω, and Pbe the probability measure on U. We define a probability spaceto be the triple (Ω, U, P).DefinitionLet (Ω, U, P) be a probability space and B be the Borel subsets ofR. Then the mappingX:Ω R(1)is a random variable if for each B B, then X 1 (B) U.

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsDifferential EquationsStochastic ODEs (cont.)DefinitionA collection {Xt t 0} of random variables is called astochastic process.HO Methods

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsDifferential EquationsStochastic ODEs (cont.)DefinitionA collection {Xt t 0} of random variables is called astochastic process.DefinitionFor each point ω Ω, the mapping t 7 Xt (ω) is thecorresponding sample path.HO Methods

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsHO MethodsDifferential EquationsStochastic ODEs (cont.)DefinitionA collection {Xt t 0} of random variables is called astochastic process.DefinitionFor each point ω Ω, the mapping t 7 Xt (ω) is thecorresponding sample path.Now we can modify the general deterministic ODE that we haveseen. Mimicking what we saw for ODEs, we write(Ẋt f (t, Xt ) F (t, Xt )ξtX0 x0for t 0x0 R

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsHO MethodsDifferential EquationsStochastic ODEs (cont.)DefinitionA collection {Xt t 0} of random variables is called astochastic process.DefinitionFor each point ω Ω, the mapping t 7 Xt (ω) is thecorresponding sample path.Now we can modify the general deterministic ODE that we haveseen. Mimicking what we saw for ODEs, we write(Ẋt f (t, Xt ) F (t, Xt )ξtX0 x0for t 0x0 Rwhere F and f are sufficiently smooth functions, and Xt is astochastic process. But what is ξt ?

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsHO MethodsDifferential EquationsWhite NoiseThe term ξt is defined to be white noise. If we rewrite the ODEwith the white noise a little bit to look like ODEs from undergradODE, we have

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsHO MethodsDifferential EquationsWhite NoiseThe term ξt is defined to be white noise. If we rewrite the ODEwith the white noise a little bit to look like ODEs from undergradODE, we havedXtdWt f (t, Xt ) F (t, Xt )dtdt(2)where Wt turns out to be Brownian motion, or a Wiener process.dmeans!) we writeSymbolically (being careful about what dtdWt ξtdt(3)

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsHO MethodsDifferential EquationsWhite NoiseThe term ξt is defined to be white noise. If we rewrite the ODEwith the white noise a little bit to look like ODEs from undergradODE, we havedXtdWt f (t, Xt ) F (t, Xt )dtdt(2)where Wt turns out to be Brownian motion, or a Wiener process.dmeans!) we writeSymbolically (being careful about what dtdWt ξtdt(3)This seems to say that the time derivative of a Brownian motion iswhite noise. We will see this is not quite correct (in the usualsense), once we define what Brownian motion is.

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsHO MethodsDifferential EquationsWhite Noise (A more formal definition)DefinitionLet T be an indexing set, and X : {Xt }t T be a stochasticprocess. Then X is a Gaussian random field (or Gaussianprocess if T R) if (Xt1 , . . . , Xtn ) is a Gaussian random vectorfor all t1 , . . . , tn T .

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsHO MethodsDifferential EquationsWhite Noise (A more formal definition)DefinitionLet T be an indexing set, and X : {Xt }t T be a stochasticprocess. Then X is a Gaussian random field (or Gaussianprocess if T R) if (Xt1 , . . . , Xtn ) is a Gaussian random vectorfor all t1 , . . . , tn T .DefinitionLet A : A(Rn ) denote the collection of all Borel-measurablesubsets of Rn that have finite Lebesgue measure. Then whitenoise on Rn is a mean-zero, set indexed, Gaussian random field{ξ(A)}A A , with covariance functionE [ξ(A1 )ξ(A2 )] : m(A1 A2 ) for all A1 , A2 Awhere m denotes Lebesgue measure.(4)

IntroductionDefs and DEsBM and SCGBMEM MethodMilstein MethodMC MethodsHO MethodsDifferential EquationsSODE in standard formReturning back to SODEs, we can write a general SODE in thegeneral differential form(dXt f (t, Xt ) dt F (t, Xt ) dWtX0 x0for t 0x0 Rwhere the terms dXt and F dWt are called stochastic differentials.